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				<h1 class="article-title-regular text-second-text-color tracking-tight text-4xl md:text-6xl font-semibold px-2 sm:px-6 md:px-8 py-3">非对称密码-RSA</h1>
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                        <a href="/tags/RSA/">RSA</a>&nbsp;
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			<p>本章内容主要为RSA密码的原理以及定位</p>
<p>分组密码在应用和协议的加解密中比较常用，非对称加密RSA因为加密效率的原因更多的用在密钥的分发方面</p>
<p>非对称密码中的RSA在当前互联网时代被广泛使用（网站、app、协议、桌面应用、服务器等等）。RSA加密算法是公钥密码体制中的一种，改算法基于数学理论而不是对称密码中的混淆和扩散。RSA密码算法当中加密和解密使用的是不同的密钥，因此被称为非对称密码。RSA是1977年由罗纳德·里维斯特、阿迪·萨莫尔和伦纳德·阿德曼一起提出的。RSA就是他们三人姓氏开头字母拼在一起的。</p>
<span id="more"></span>

<p>RSA原理来自数论相关知识。RSA公开密钥密码体制的原理是：根据数论，寻求两个大素数比较简单，而将它们的乘积进行因式分解却极其困难，因此可以将乘积公开作为加密密钥。</p>
<blockquote>
<p>找素数</p>
<ul>
<li>选取两个大的随机素数</li>
</ul>
<p>计算模n和Euler函数<strong>φ(n)</strong></p>
<ul>
<li>$n = pq$</li>
<li>$φ(n) = (p-1)(q-1)$</li>
</ul>
<p>找$ed ≡ 1 mod φ(n)$</p>
<ul>
<li>随机取一个数e（与φ(n)互素），用扩展Euclid算法求d即可</li>
</ul>
<p>发布</p>
<ul>
<li>d保密，（d，n）是私钥$K_s$</li>
<li>发布（e，n），这是公钥$K_p$</li>
<li>销毁p、q</li>
</ul>
<p>加密</p>
<p>​	$c = m^e ,mod ,n$</p>
<p>解密</p>
<p>​	$m = c^d,mod,n$</p>
<p>明文分组m为整数且必须小于n</p>
</blockquote>
<h6 id="消息发送的流程"><a href="#消息发送的流程" class="headerlink" title="消息发送的流程"></a>消息发送的流程</h6><img lazyload="" src="/images/loading.svg" data-src="/2024/03/05/%E9%9D%9E%E5%AF%B9%E7%A7%B0%E5%AF%86%E7%A0%81-RSA/image-20240305110855142.png" class="" title="image-20240305110855142">

<blockquote>
<ol>
<li><p>随机选择两个不相等的质数p和q例如61和53</p>
</li>
<li><p>计算p和q的乘积n，$n = 61 ✖️ 53 = 3233$，3233写成二进制是110010100001，一共有12位，即这个密钥就是12位</p>
</li>
<li><p>计算n的欧拉函数$φ(n)$</p>
<p>根据公式：$φ(n) = (p-1)(q-1)$ 算出$φ(3233)$等于60✖️52，即3120</p>
</li>
<li><p>选择一个整数e，条件是$1 &lt; e &lt; φ(n)$，且e与φ(n)互质。在1到3120之间，随机选择了17（实际应用中，常常选择65537）</p>
</li>
<li><p>计算e对于φ(n)的模反元素d</p>
<p>所谓“模反元素”就是指有一个整数d，可以使得ed被φ(n)除的余数为1</p>
<p>​	$ed ≡ 1(mod,φ(n))$</p>
<p>​	等价于</p>
<p>​	$ed -1 = kφ(n)$</p>
<p>于是，找到模反元素d，实质上就是对下面这个二元一次方程求解</p>
<p>​	$ex + φ(n)y = 1$</p>
<p>已知$e = 17$ $φ(n) = 3120$</p>
<p>​		$17x + 3120y = 1$</p>
<p>这个方程可以用“扩展欧几里得算法”求解，此处省略具体过程。总之，算出一组整数解为$(x, y) = (2753, -15)$，即$d = 2753$</p>
</li>
<li><p>将n和e封装成公钥，n和d封装成私钥</p>
<p>例如：n = 3233，e = 17， d = 2753，所以公钥就是（3233， 17），私钥就是（3233， 2753）。有了公钥和私钥，就能进行加密和解密了。</p>
</li>
</ol>
<p>下面具体过程可以参考<a class="link" target="_blank" rel="noopener" href="https://blog.csdn.net/lu1196700392/article/details/102593536">这篇帖子<i class="fa-solid fa-arrow-up-right ml-[0.2em] font-light align-text-top text-[0.7em] link-icon"></i></a></p>
</blockquote>
<h6 id="安全性"><a href="#安全性" class="headerlink" title="安全性"></a>安全性</h6><p>对极大整数做因式分解的难度决定了RSA算法的可靠性。到目前为止公开的被破解的最长RSA密钥是768位。实际应用中一般情况下用1024位，重要场合2048位还都是安全的。</p>
<h6 id="如何加密长信息"><a href="#如何加密长信息" class="headerlink" title="如何加密长信息"></a>如何加密长信息</h6><p>公钥（n，e）只能加密小于n的整数m，那么如果要加密大于n的整数，该怎么办？有两种解决方法：一种是把长信息分割成若干段短消息，每段分别加密；另一种是先选择一种“对称加密算法（例如：DES）”，用这种算法的密钥加密信息，再用RSA公钥加密DES密钥</p>
<p>因为RSA的加密效率原因，通常会采用第二种方式</p>
<h6 id="RSA中的填充"><a href="#RSA中的填充" class="headerlink" title="RSA中的填充"></a>RSA中的填充</h6><p>在之前的分组加密中，Padding只是用来填充数据到指定的长度，而RSA通常不会加密特别长的数据，因此没有分组模式的概念，对于RSA来说Padding是既有分组模式的概念又有随机数的概念。即RSA的Padding包含了将数据填充到RSA密钥位数的长度的方法，还有填充随机数到RSA原文的方法</p>
<p>RSA算法的本质就是大数运算</p>
<p>​	$m^e ≡ c(mod\ n)$</p>
<p>​	$c^d ≡ c(mod\ n)$</p>
<p>其中m是原文，c是密文。如果使用原始的RSA做加解密操作，则并不包含随机数，相同的密文会生成相同的明文。同时由于大部分情况下m相比n小很多，甚至$m^e$都比n要小，这时候很容易通过枚举倍数破解明文。因此RSA加解密算法很需要有效的Padding算法讲明文填充到足够长保证不容易被暴力破解，这也就是加入随机因子保证密文的随机性的一个原因。</p>
<h6 id="查看RSA密钥"><a href="#查看RSA密钥" class="headerlink" title="查看RSA密钥"></a>查看RSA密钥</h6><p>用openssl生成一个RSA的key来看一下</p>
<p>生成</p>
<p><code>openssl genrsa -out pv.key 1024</code></p>
<p>从私钥中获取p、q、n、e、d信息</p>
<img lazyload="" src="/images/loading.svg" data-src="/2024/03/05/%E9%9D%9E%E5%AF%B9%E7%A7%B0%E5%AF%86%E7%A0%81-RSA/image-20240305162959247.png" class="" title="image-20240305162959247">

<h6 id="实现RSA"><a href="#实现RSA" class="headerlink" title="实现RSA"></a>实现RSA</h6><p>在C/C++中使用RSA：不得不提的OpenSSL</p>
<p>OpenSSL开源库中针对当前流行密码学领域相关算法都进行了实现，其中就包含有非对称密码中的RSA算法。对于开发人员来说，很难去自己实现一个兼顾鲁棒性和效率的RSA加解密代码，因此也就一般都直接使用OpenSSL开源库。Google创建了OpenSSL分支：BoringSSL，但Google不打算取代OpenSSL，使用BoringSSL的代码不能保证API或ABI的稳定性，他们会继续向OpenSSL递交bug修正，继续资助相关的开源基金会。</p>
<h6 id="hook和定位RSA"><a href="#hook和定位RSA" class="headerlink" title="hook和定位RSA"></a>hook和定位RSA</h6><p>对于Java层的RSA还是和之前DES、AES一样，都是调用cipher实现的，所以还是同样的hook方式</p>
<p>对于so层来说，它的实现基本都是直接使用openssl的公开库，直接调用的。因此对于它的识别只需要使用ida插件快速定位到openssl库就可以了</p>
<p>定位RSA算法主要有：识别常量表、控制流、重放、识别OpenSSL库的版本制作签名库几种方法。</p>

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